A paradox of solid state physics is that electrons in conductors are almost exclusively regarded as non-interacting particles, even though they do interact. This comes both from physical reasons and from the human need for convenience. The physical reason is that the interacting electrons form a ground state, and charged elementary excitations above the state – quasielectrons – do not interact provided their energies are sufficiently close to the Fermi surface. This makes a model of non-interacting electrons completely adequate for quasielectrons, at least in the low-energy limit. This allows a scattering approach to quantum transport that assumes the absence of interaction. The convenience model is that the physics of non-interacting particles is much easier to understand and apply. Besides, sticking to a convenient picture usually goes unpunished. In fact, in solid state physics there are only a few rather exotic examples where interaction effects really reign and the noninteracting approach produces obviously erroneous results. These cases are notoriously difficult to comprehend and to quantify; some effects revealed almost a century ago (for instance, Mott insulator transition) are still on the front-line of modern research.

In contrast to solid state physics, there is a very common regime in quantum transport where interaction effects are dominant: the Coulomb blockade regime, and here the scattering approach fails. However, in contrast to the situation in solid state systems, Coulomb blockade systems are usually even simpler than those of Chapters 1 and 2 where interaction does not play an important role. Due to this, one can quickly grasp the fascinating features of Coulomb blockade physics, and begin to design and apply Coulomb blockade circuits and devices.

It is an interesting intellectual game to compress an essence of a science, or a given scientific field, to a single sentence. For natural sciences in general, this sentence would probably read: Everything consists of atoms. This idea seems evident to us. We tend to forget that the idea is rather old: it was put forward in Ancient Greece by Leucippus and Democritus, and developed by Epicurus, more than 2000 years ago. For most of this time, the idea remained a theoretical suggestion. It was experimentally confirmed and established as a common point of view only about 150 years ago.

Those 150 years of research in atoms have recently brought about the field of nanoscience, aiming at establishing control and making useful things at the atomic scale. It represents the common effort of researchers with backgrounds in physics, chemistry, biology, material science, and engineering, and contains a significant technological component. It is technology that allows us to work at small spatial scales. The ultimate goal of nanoscience is to find means to build up useful artificial devices – nanostructures – atomby atom. The benefits and great prospects of this goal would be obvious even to Democritus and Epicurus.

This book is devoted to quantum transport, which is a distinct field of science. It is also a part of nanoscience. However, it is a very unusual part. If we try to play the same game of putting the essence of quantum transport into one sentence, it would read: It is not important whether a nanostructure consists of atoms.

We devoted Chapter 1 to a purely quantum-mechanical approach to electron transport: the scattering approach. Electrons were treated as quantum waves that propagate between reservoirs – the contact pads of a nanostructure. The waves experience scattering, and the transport properties are determined by the scattering matrix of these waves. As we have seen, this approach becomes progressively impractical with the increasing number of transport channels, and can rarely be applied for G ≫ GQ, where G is the conductance of the system.

A different starting point is well known from general physics, or, more simply, from general life experience, which is rather classical. In this context, a nanostructure is regarded as an element of an electric circuit, which conducts electric currents. If one makes a more complicated circuit by combining these elements, one does not have to involve quantum mechanics to figure out the result. Rather, one uses Ohm's law or, generally, Kirchhoff rules. The number of parameters required for this description is fewer than in the quantum-mechanical scattering approach. For example, the phase shifts of the scattering matrix do not matter.

In this chapter, we will bridge the gap between these opposite starting points. The first bridge is rather obvious: it is important to understand that these two opposite approaches do not contradict each other. In Section 2.1, we illustrate the difference and the link between the approaches with a comprehensive example of a double-junction nanostructure.

This book provides an introduction to the rapidly developing field of quantum transport. Quantum transport is an essential and intellectually challenging part of nanoscience; it comprises a major research and technological effort aimed at the control of matter and device fabrication at small spatial scales. The book is based on the master course that has been given by the authors at Delft University of Technology since 2002. Most of the material is at master student level (comparable to the first years of graduate studies in the USA). The book can be used as a textbook: it contains exercises and control questions. The program of the course, reading schemes, and education-related practical information can be found at our website www.hbar-transport.org.

We believe that the field is mature enough to have its concepts – the key principles that are equally important for theorists and for experimentalists – taught. We present at a comprehensive level a number of experiments that have laid the foundations of the field, skipping the details of the experimental techniques, however interesting and important they are. To draw an analogy with a modern course in electromagnetism, it will discuss the notions of electric and magnetic field rather than the techniques of coil winding and electric isolation.

We also intended to make the book useful for Ph.D. students and researchers, including experts in the field. We can liken the vast and diverse field of quantum transport to a mountain range with several high peaks, a number of smaller mountains in between, and many hills filling the space around the mountains.

A Second Edition has given me several opportunities, which I have grasped with some eagerness. The first involved a matter of house-keeping – the correction of typos and an equation (Eq. (11.31)) in the First Edition. The second has allowed me to add the theory of the elasticity of optical modes, which became formulated too late to be included in the First Edition, but which resolved some uncertainties regarding mechanical boundary conditions. Given the current interest in hot phonons in high-power devices, I have updated the section on phonon lifetime. But the greatest opportunity was to expand the book to include some of the topics that grew in significance during the decade following the writing of the First Edition. Four new chapters focus on spin relaxation, the III–V nitrides, and the generation of terahertz radiation. In the burgeoning technology of spintronics, the rate of decay of the spin-polarization of the electron gas is a crucial parameter, and I have reviewed the mechanisms for this in both bulk and low-dimensional material. The advance of growth techniques and the technological need for higher-powered devices and for visible LEDs have thrust AIN, GaN, and InN and their alloys into the forefront of semiconductor physics. New properties associated with the hexagonal lattice have presented a challenge, and these are described in the chapter on electrons and phonons in the wurtzite lattice, and their role in heterostructures and multilayers is reviewed in a further chapter.

General Remarks

Evidently, the advent of mesoscopic layered semiconductor structures generated a need for a simple analytic description of the confinement of electrons and phonons within a layer and of how that confinement affected their mutual interaction. The difficulties encountered in the creation of a reliable description of excitations of one sort or another in layered material are familiar in many branches of physics. They are to do with boundary conditions. The usual treatment of electrons, phonons, plasmons, excitons, etc., in homogeneous bulk crystals simply breaks down when there is an interface separating materials with different properties. Attempts to fit bulk solutions across such an interface using simple, physically plausible connection rules are not always valid. How useful these rules are can be assessed only by an approach that obtains solutions of the relevant equations of motion in the presence of an interface, and there are two types of such an approach. One is to compute the microscopic band structure and lattice dynamics numerically; the other is to use a macroscopic model of longwavelength excitations spanning the interface. The latter is particularly appropriate for generating physical concepts of general applicability. Examples are the quasi-continuum approach of Kunin (1982) for elastic waves, the envelope-function method of Burt (1988) for electrons and the wavevector-space model of Chen and Nelson (1993) for electromagnetic waves and excitons.

Charged-Impurity Scattering

Introduction

Scattering of electrons by charged-impurity atoms dominates the mobility at low temperatures in bulk material and is usually very significant at room temperature (Fig. 9.1). The technique of modulation doping in high-electron-mobility field-effect transistors (HEMTs) alleviates the effect of charged-impurity scattering but by no means eliminates it. It remains an important source of momentum relaxation (but not of energy relaxation because the collisions are essentially elastic). Though its importance has been recognized for a very long time, obtaining a reliable theoretical description has proved to be extremely difficult.

There are many problems. First of all there is the problem of the infinite range of the Coulomb potential surrounding a charge, which implies that an electron is scattered by a charged impurity however remote, leading to an infinite scattering cross-section for vanishingly small scattering angles. Intuitively, we would expect distant interactions with a population of charged impurities to time-average to zero, leaving only the less frequent, close collisions to determine the effective scattering rate. This intuition motivated the treatment by Conwell and Weisskopf (1950) in which the range of the Coulomb potential was limited to a radius equal to half the mean distance apart of the impurities. Setting an arbitrary limit of this sort was avoided by introducing the effect of screening by the population of mobile electrons as was done by Brooks and Herring (1951) for semiconductors, following the earlier approach by Mott (1936).

Introduction

In recent years there has been growing interest in sources of radiation whose frequencies occupied the region of the electromagnetic spectrum between, roughly, 300GHz and 30THz. The numerous applications include their use in medicine, molecular spectroscopy, communication, and security. The frequency range is covered by free-electron lasers, but there is an obvious need for more portable and adaptable sources, and this need has focussed attention on the properties of semiconductors and semiconductor multilayers. Existing semiconductor microwave generators such as the Gunn diode run out of power above 100GHz and IMPATTs above 300GHz. A new generation of devices that can extend the wavelength range beyond the familiar millimetre wave regime and into the sub-millimetre regime is required, and there has been substantial progress in recent years, much of it associated with the exploitation of the femtosecond Ti-sapphire laser and various non-linear optical processes. Semiconductors have also been used as THz detectors, notably, Ge and Si bolometers operating at liquid helium temperatures. The photon energy at 1THz is 4meV, and photon energies of room temperature radiation are around 25meV, so the problem of detection of THz radiation is that it is always against a competing, pervading background. Mention in Section 14.6 has already been made of cascade lasers operating in the near infrared.

The Wurtzite Lattice

Wurtzite is zinc oxide. Its lattice is made up of tetrahedrally bonded atoms, as it is in zinc blende, but it differs in the stacking sequence: in zinc blende the structure has face-centred cubic symmetry, in wurtzite the structure has hexagonal symmetry. The wurtzite lattice is the preferred thermodynamically stable phase of a number of binary semiconductors, among them the nitrides, AlN, GaN, and InN, which have become of considerable technical importance in recent years. Among the applications are high-power microwave transistors, light-emitting diodes, and highly efficient solar cells. If the paradigm of the cubic III-V semiconductors is GaAs, the paradigm of the hexagonal nitrides is GaN. Being familiar with the properties of GaAs, we need to know how the lower symmetry of GaN affects the band structure of electrons and holes, the phonon spectrum and the electron–phonon interaction. We focus on these topics in this chapter. Fuller accounts of the properties of the nitrides can be found in a number of excellent reviews (e.g. Ambacher, 1998; Ambacher et al., 2002; Piprek (ed.) Nitride Semiconductor Devices, 2007).

The unit cell of the wurtzite lattice is depicted in Fig. 13.1. It consists of stacked hexagonal planes so that were the atoms to be spherical with the same radius, the lattice would be a close-packed hexagonal structure.

Introduction

We now turn to the problem of describing long-wavelength lattice vibrations in multilayered structures. At a microstructural level this problem is solved by using the techniques of lattice dynamics requiring intense numerical computation. This approach is inconvenient, if not impracticable, at the macroscopic level, where the kinetics and dynamics of large numbers of particles need to be described. In this theoretical regime it is necessary to obtain models that transcend those at the level of individual atoms in order to describe electron and hole scattering, and all the energy and momentum relaxation processes that underlie transport and optical properties of macroscopic structures. The basic problem is to make a bridge between the atomic crystal lattice and the classical continuum. We know from the theory of elasticity and the theory of acoustic waves, which hark back to the nineteenth century, that continuum theory works extremely well for long-wave acoustic waves. The case of optical vibrations is another matter. Here, the essence is one atom vibrating against another in a primitive unit cell and it is by no means obvious that a continuum approach can work in this case. This has been highlighted by controversy concerning the boundary conditions that long-wave optical vibrations obey at each interface of a multilayer structure.